(1) Field of the Invention
The present invention relates generally to signal prediction and more particularly to a method and apparatus for signal prediction in a time-varying signal system.
(2) Description of the Prior Art
Currently, the cancellation of time-varying noise signals has been made possible by a breakthrough in the field of active noise control. Specifically, applicant's recently filed U.S. patent application, copending Ser. No. 07/573415, incorporated herein by reference, teaches the combining of open and closed-loop responses to cancel the time-varying noise signal. The approach models the noise field (from both on-line data generated from the actual noise field, and off-line data from a historical data base) at any point as a stochastic process. To adaptively estimate the characteristics of this process, one of several algorithms may be used. One such algorithm is the estimate-maximize (EM) algorithm.
The EM algorithm iteratively obtains a Maximum Likelihood (ML) estimate of the unknown parameters using the notion of complete and incomplete date sets. The ML estimation is regarded as the optimal method for parameter estimation. Given a set of observed (incomplete) data z, the ML estimate of the vector of unknown parameters .theta. is defined as EQU .theta..sub.ML =arg.sub..theta. max log f.sub.z (z;.theta.) (1)
where log f.sub.z (z;.theta.) is the logarithm of the likelihood function of z, and f.sub.z (z;.theta.) is the probability density function of z for a given set of parameters .theta.. Because the parameter vector .theta. contains several unknowns and log f.sub.z (z;.theta.) is generally a nonlinear function of .theta., the maximization of equation (1) tends-to be very complex.
Accordingly, the EM algorithm is used to find the ML estimate based on complete and incomplete data sets. The observed data set z is treated as the incomplete data while the complete data set y is such that: EQU z=H(y) (2)
where H is a non-invertible (many-to-one) transformation. The EM algorithm is an iterative method that starts with an initial guess .theta..sup.0, and then inductively calculates .theta..sup.L in two steps, namely, the estimate step (E-step) and the maximize step (M-step) defined as follows: ##EQU1##
The EM algorithm is not uniquely defined since the transformation H relating the complete data set y to the incomplete data set z can be any non-invertible transformation. Thus, there are many possible complete data specifications that will generate the incomplete (observed) data. However, H should be chosen such that the M-step is computationally simple thereby reducing the time for each iteration. At the same time, the resulting complete data must be sufficiently correlated with the incomplete data to guarantee a fast rate of convergence.
The rate of convergence of the EM algorithm depends on the cross-correlation or covariance of the complete data with the incomplete data. Also, the majority of the time required for solution convergence lies in the first maximization step. Accordingly, the speed of convergence for the complete solution depends greatly on the value of the initial estimate .theta..sup.0.